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# Computational number theory

## What is computational number theory?

Computational number theory, also known as algorithmic number theory is a branch of number theory. It focuses on identifying and using efficient computational methods and algorithms to solve multiple problems in number theory as well as arithmetic geometry.

In recent years, there has been a lot of progress in this sphere, in terms of higher computational speed as well as the identification of increasingly efficient algorithms.

Computational number theory is used heavily for primality testing as well as the prime factorization of large integers.

It is also used for the purpose of searching for solutions to diophantine equations, and for explicit methods in arithmetic geometry.

Computational number theory is used widely in cryptography. It is used in elliptic curve cryptography, RSA, and post-quantum cryptography. The theory is even used to investigate speculations, notions, and open problems in number theory.

## What is primality testing?

Primality testing determining whether a number is a prime number or not, without decomposing it into its constituent prime factors. There are two types of primality testing:

• Deterministic primality testing: This gives you an absolutely certain answer about the primality of a number.
• Probabilistic primality testing: Here it is possible (with a minuscule probability) to falsely identify a composite number as a prime number.

## What is the prime factorization of integers?

Prime factorization is the process of determining the prime factors of a number. This is considered to be a computationally hard problem and there are multiple algorithms with varying levels of sophistication that are used to solve this problem.

Direct search factorization is the simplest way to find the factors. It uses trial division of all possible factors. However, it is only practical for small numbers. Computational number theory is useful for the prime factorization of larger numbers.

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# Computational number theory

October 14, 2020

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## What is computational number theory?

Computational number theory, also known as algorithmic number theory is a branch of number theory. It focuses on identifying and using efficient computational methods and algorithms to solve multiple problems in number theory as well as arithmetic geometry.

In recent years, there has been a lot of progress in this sphere, in terms of higher computational speed as well as the identification of increasingly efficient algorithms.

Computational number theory is used heavily for primality testing as well as the prime factorization of large integers.

It is also used for the purpose of searching for solutions to diophantine equations, and for explicit methods in arithmetic geometry.

Computational number theory is used widely in cryptography. It is used in elliptic curve cryptography, RSA, and post-quantum cryptography. The theory is even used to investigate speculations, notions, and open problems in number theory.

## What is primality testing?

Primality testing determining whether a number is a prime number or not, without decomposing it into its constituent prime factors. There are two types of primality testing:

• Deterministic primality testing: This gives you an absolutely certain answer about the primality of a number.
• Probabilistic primality testing: Here it is possible (with a minuscule probability) to falsely identify a composite number as a prime number.

## What is the prime factorization of integers?

Prime factorization is the process of determining the prime factors of a number. This is considered to be a computationally hard problem and there are multiple algorithms with varying levels of sophistication that are used to solve this problem.

Direct search factorization is the simplest way to find the factors. It uses trial division of all possible factors. However, it is only practical for small numbers. Computational number theory is useful for the prime factorization of larger numbers.